Quick Review: Quadrilaterals and Polygons

Quadrilateral
A simple closed four sided figure is known as a Quadrilateral.
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Types of Quadrilaterals:
  • Parallelogram: In a parallelogram, opposite sides are parallel and congruent.
    Area = length * height
    Perimeter = 2 (length + breadth)
  • Rectangle: In a rectangle, opposite sides are parallel and congruent.
    Area = length * breadth
    Diagonal of the rectangle = √(l2 + b2)
    Area = length * breadth
    Perimeter = 2 (length + breadth)
  • Square: All sides and angles of a square are congruent.
    Length of the diagonal = l √2 (where l is the length of the side of a square)
    Area = (Side)2
    Perimeter = 4 X side
  • Rhombus: All sides and all opposite angles of a rhombus are congruent. Also, the diagonals are perpendicular to and each other.
    Area = (a*b)/2 (where a and b are the lengths of the diagonals of a rhombus)
    Perimeter = 4 * length
  • Trapezium: No sides, angles and diagonals of a trapezium are parallel to each other.
    Area = (1/2)h*(L+L2)
    Perimeter = L + L1 + L2 + L3
Important Points
  • Of all the quadrilaterals of the same perimeter, the one with the maximum area is the square.
  • The quadrilateral formed by joining the midpoints of the sides of any quadrilateral is always a parallelogram (rhombus in case of a rectangle, rectangle in case of a rhombus and square in case of a square).
  • The quadrilateral formed by the angle bisectors of the angles of a parallelogram is a rectangle.
  • For a rhombus □ABCD, if the diagonals are AC and BD, then AC2 + BD2 = 4 * AB2.
  • If a square is formed by joining the midpoints of a square, then the side of the smaller square = side of the bigger square/√2.
  • If P is a point inside a rectangle □ABCD, then AP2 + CP2 = BP2 + DP2.
  • The segment joining the midpoints of the non-parallel sides of a trapezium is parallel to the two parallel sides and is half the sum of the parallel sides.
  • For a trapezium □ABCD, if the diagonals are AC and BD, and AB and CD are the parallel sides, then AC2 + BD2 = AD2 + BC2 + 2 * AB * BD.
  • If the length of the sides of a cyclic quadrilaterals area, b, c and d, then it's area = √(s-a)(s-b)(s-c)(s-d) , where s is the semi - perimeter.
  • The opposite angles of a cyclic quadrilateral are supplementary.
Polygons
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Properties
  • The sides an of regular inscribed polygons, where R is the radius of the circumscribed circle = an = 2R sin 180o/n
  • Area of a polygon of perimeter P and radius of in-circle r = (P*r)/2
  • The sum of the interior angles of a convex POLYGON, having n sides is 180o (n - 2).
  • The sum of the exterior angles of a convex polygon, taken one at each vertex, is 360o.
  • The measure of an exterior angle of a regular n- sided polygon is 360o/n .
  • The measure of the interior angle of a regular n-sided polygon is (n-2)180o/n
  • The number of diagonals of in an n-sided polygon is n(n-3)/2